Reddit mentions of Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd Edition

Well... this may be the book you're looking for.

After a brief review of linear algebra and differential calculus, it very methodically builds up matrix calculus from foundational principles, taking a very abstract algebra/metric space topology approach. Be warned though... this is a serious fucking book. When I say you should be comfortable with linear algebra, I don't mean you know how to think in terms of basis vectors and eigenvalues and maybe some spectral theory and quadratic forms. I mean you need to be very comfortable with some fairly advanced matrix algebra, especially dealing with partitioned matrices and determinant calculations. The calculus foundation comes at it from a functional analysis perspective, and will be much easier if you've already gotten something of a foundation in analysis, and ideally some metric theory. If you can handle it, this is exactly what you're looking for. There's quite a few exercises and examples, everything is rigorously proven... what I've been able to get through of the book so far has been useful. But caveat emptor... I shelved it partway in, I'm planning on looping back in a year after I get through Pugh's analysis book and Roman's linear algebra book. This book is not for the faint of heart.

My suggestion... see if you can find a pdf or something and try and go through the first section or two in both the first and the second chapter. The first two chapters are largely review, and aren't intended to be instructional... more like a brief reminder before jumping into the deep end. If those two chapters aren't mostly review, then you need to work your way up to this book with some supplementary resources.